Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice

El Haddad Rami (), Lécot Christian () and Venkiteswaran Gopalakrishnan ()
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El Haddad Rami: Département de Mathématiques, Faculté des Sciences, Université Saint-Joseph, BP 11-514 Riad El Solh, Beyrouth 1107 2050, Lebanon. E-mail:
Lécot Christian: Laboratoire de Mathématiques, UMR 5127 CNRS and Université de Savoie, Campus scientifique, 73376 Le Bourget-du-Lac Cedex, France. E-mail:
Venkiteswaran Gopalakrishnan: Department of Mathematics, Birla Institute of Technology and Science, Vidya Vihar Campus, Pilani, 333 031 Rajasthan, India. E-mail:

Monte Carlo Methods and Applications, 2010, vol. 16, issue 3-4, 211-230

Abstract: We are interested in Monte Carlo (MC) methods for solving the diffusion equation: in the case of a constant diffusion coefficient, the solution is approximated by using particles and in every time step, a constant stepsize is added to or subtracted from the coordinates of each particle with equal probability. For a spatially dependent diffusion coefficient, the naive extension of the previous method using a spatially variable stepsize introduces a systematic error: particles migrate in the directions of decreasing diffusivity. A correction of stepsizes and stepping probabilities has recently been proposed and the numerical tests have given satisfactory results. In this paper, we describe a quasi-Monte Carlo (QMC) method for solving the diffusion equation in a spatially nonhomogeneous medium: we replace the random samples in the corrected MC scheme by low-discrepancy point sets. In order to make a proper use of the better uniformity of these point sets, the particles are reordered according to their successive coordinates at each time step. We illustrate the method with numerical examples: in dimensions 1 and 2, we show that the QMC approach leads to improved accuracy when compared with the original MC method using the same number of particles.

Keywords: Quasi-Monte Carlo; random walk; low-discrepancy sequences; diffusion equation (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1515/mcma.2010.009

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